In chapter 23, there is a passage where Scout notices that Jem is changing in appearances lately. She notices that
"his eyebrows were becoming heavier, and...a new slimness about his body. He was growing taller."
Then, Jem proudly shows Scout his first chest hair, and tells her that he's "goin' out for football next year" also. So, there are the physical changes that are mentioned in that chapter. He is hitting a growth spurt, and starting the track to becoming more of a man in appearances.
Other ways that Jem "comes of age" throughout the novel are that he understands intricate human situations, like how his dad doesn't want to brag about being a good shot, and how, even though it upsets him, people can make the wrong decisions out of fear. He is processing the world and figuring things out for himself. He even analyzes people in Maycomb, and figures out that what separates people from each other is how long they've "been readin' and writin'", and concludes with some wisdom:
"I think I'm beginning to understand why Boo Radley's stayed shut up in the house all this time...it's because he wants to stay inside."
Jem has figured out that the world isn't an ideal place where everything fits nicely; he is losing that childhood naivety where kids think everything is happy and perfect. He is coming of age in the sense that he is developing a more realistic view of people and the world around him. I hope that those thoughts help; good luck!
Jem starts to understand the world a bit more and know that the world is not as innocent as it seems. When Jem receives Atticus' pocketwatch it symbolizes a new change and growth in him as he is maturing into a young man. He learns not to play as a child anymore but to think more maturely.
There are many examples of this the physical changes of puberty he goes through also Jem becomes less naive and innocent of teh world.
Anotehr example is how Jem is allowed to have Atticus' pocketwatch every so often it is a great responsibilty and symbolises age and time. He finds winding it a nuisance sometimes and as adults tehre are unwanted burdens